Infinity

Originally posted by FabianFnas
Okay, usual questions about infinities are:
(And the answers are quite contra-intiutive.)

(1) There are infinitly many natural numbers (positive whole numbers). Right?
(2) But if you bring in even the negative whole numbers into that, there should be the double amount of numbers, right?
(3) How many more (double?, triple?) are there rational nubers ...[text shortened]... uestion:
(8) How many more squares (from the natural numbers) is there compared with the qubes?

Okay, I think I should expand on what it means for two sets to have the same cardinality. Basically, it means that there exists a bijection between the two sets. This is obvious for finite cases.

{1,2,3} and {3,4,5} have the same cardinality - the mapping that sends 1->3, 2->4 and 3->5 is obviously a bijection.

However, it is also true for infinite cases. Well, not soo much true, it is by definition that two infinite sets have the same cardinality if and only if there exists a bijection between them. Also, if there exists an injection from A to B we can deduce that the cardinality of A is less than or equal to the cardinality of B. It follows that it there also exists an injection from B to A then their cardinalities are equal, and so there must exist a bijection from A to B.

For instance, the mapping from the natural numbers to the integers,
a=1: a->0
a even: a->-a/2
a odd: a->(a-1)/2
Is a bijection. Thus, the natural numbers have the same cardinality as the integers.

Also, if a set has cardinality less than or equal to the natural numbers it is said to be "countable".

I hadn't posted this in my previous post as I've no idea if you know what a bijection is...

Now, to answer your questions:

1. Correct
2. Correct, intuitively. However, by our definition they have the same cardinality
3. Here we have something of the form NxN - the set of all pairs of natural numbers (As N=Z so we can simplify it as it is actually NxZ). There is a bijection from NxN to N, but it is a bit too complicated to write here...
4. Yes, there is a lot more real numbers than natural number. Their cardinalities are completly different. |R|=2^|N| as we can show there exists a bijection from the powerset of the natural numbers to R, and also that the cardinality of the powerset of any set is strictly greater than the cardinality of that set, even in infinite cases.
5. There is uncountably many. That is, there is strictly more in that tiny interval than there is natural numbers.
6. I have no idea.
7. The cardianlity of the Real numbers is strictly greater than that of the rational numbers...

Yes, I believe a lot of these results stem from work by Cantor. Although I'm not entirely sure. 😛

Originally posted by twhitehead
Actually they are of the same cardinality.
It is incorrect to say they are the same number as infinity is not a number.
It is true to say that you can create a one-to-one mapping between the two sets, but again it is incorrect to say that they have the same number - because there is no number that represents the number of elements in the set!
The sets ...[text shortened]... are:" then the sequence must have a final element which requires that the sequence is finite.

huh?

how about a quick copy and paste from wiki ...
"In mathematics, cardinal numbers, or cardinals for short, are generalized numbers used to measure the cardinality (size) of sets. For finite sets, the cardinality is given by a natural number, which is simply the number of elements in the set. There are also transfinite cardinal numbers that describe the sizes of infinite sets."

did you see "number" written anywhere there?

Originally posted by Swlabr
Okay, I think I should expand on what it means for two sets to have the same cardinality. Basically, it means that there exists a bijection between the two sets. This is obvious for finite cases.

{1,2,3} and {3,4,5} have the same cardinality - the mapping that sends 1->3, 2->4 and 3->5 is obviously a bijection.

However, it is also true for infinite case believe a lot of these results stem from work by Cantor. Although I'm not entirely sure. 😛

number 6 ... the hardest part is proving it is possible to find a rational in between any two reals (rationals dense in reals) ... once you have that it simply becomes countable.

Originally posted by flexmore
huh?

how about a quick copy and paste from wiki ...
"In mathematics, cardinal numbers, or cardinals for short, are generalized numbers used to measure the cardinality (size) of sets. For finite sets, the cardinality is given by a natural number, which is simply the number of elements in the set. There are also transfinite cardinal numbers that describe the sizes of infinite sets."

did you see "number" written anywhere there?

But it is deceptive as the 'number' in question is not a 'natural number' (as is made clear in your quote) ie it is not a member of the set of natural numbers. Therefore we cannot apply rules like addition, subtraction, multiplication etc on it when those rules are defined as being specifically on natural numbers.

I personally think it is poor use of language to call it a number and that it is bound to lead to confusion if one does so. When we say 'number' we almost always think of a natural number, real number or in the appropriate context a complex number but 'infinity' of any cardinality is not a member of any of those sets.

Originally posted by twhitehead
But it is deceptive as the 'number' in question is not a 'natural number' (as is made clear in your quote) ie it is not a member of the set of natural numbers. Therefore we cannot apply rules like addition, subtraction, multiplication etc on it when those rules are defined as being specifically on natural numbers.

I personally think it is poor use of l a complex number but 'infinity' of any cardinality is not a member of any of those sets.

most people think of 1 2 3 etc as numbers .. common sense

and then there is some infinite thing ... when normal people try to think of infinite as one number it all falls apart ... and so it should ... infinite is not one number ....

infinite is a set of numbers. there are different types of infinite (or cardinalities or whatever you want to call them) ... but in my mind they certainly are numbers every bit as real and defined as 1, 2 and 3.

why are they numbers? because the number system is incredibly flimsy and weak without them ... it is not complete without them. ... the number system needs them ... they are part of the number system.

i would like someone to explain the usefulness of infinity. sure it is a concept designed to explain some concepts as to what happens when you divide 1 by an increasingly higher number(nearing infinity).
(as a person who hates math, i find it simply a concept invented by mathematicians because they decided it would be more productive playing with numbers than rotting in front of a TV)


but as to real life applications who can give a formula that uses infinity?

Originally posted by whodey
Absolutly right. You cannot add nor subtract infinity. You can only drive it. 😛

You can only drive through it. Infinity is a mad abstract mathematical concept, it's fun to play around with.

Originally posted by Zahlanzi
i would like someone to explain the usefulness of infinity. sure it is a concept designed to explain some concepts as to what happens when you divide 1 by an increasingly higher number(nearing infinity).
(as a person who hates math, i find it simply a concept invented by mathematicians because they decided it would be more productive playing with numbers ...[text shortened]... front of a TV)


but as to real life applications who can give a formula that uses infinity?

Here is the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = amount after time "t"
P = principal (amount at t=0)
r = interest rate
n = number compounding periods

At one time, it was thought possible to earn unlimited amounts of money by increasing "n", thereby accumulating smaller bits of interest more rapidly. However, by letting "n" approach infinity, it was discovered that this formula is limited as follows:

A = P * exp(rt)

Where "exp" is the exponential function. This is why the bank can't take all your money instantly every time you take out a loan, even though they try their darndest. 😉

Originally posted by flexmore
number 6 ... the hardest part is proving it is possible to find a rational in between any two reals (rationals dense in reals) ... once you have that it simply becomes countable.

But surely that cannot be true - The real numbers are uncountable, while the rationals are countable. Thus, the compliment of the reals with the rationals is uncountable (uncountable-countable=uncountable). Thus, if you can find a countable number of reals between every pair of rational numbers the compliment is countable, contradiction...

But as for the other way round? if you can find a rational between any two irrational numbers then there cannot be more reals than rationals, thus contradiction...?

Originally posted by Swlabr
But surely that cannot be true - The real numbers are uncountable, while the rationals are countable. Thus, the compliment of the reals with the rationals is uncountable (uncountable-countable=uncountable). Thus, if you can find a countable number of reals between every pair of rational numbers the compliment is countable, contradiction...

But as for the o ...[text shortened]... ny two irrational numbers then there cannot be more reals than rationals, thus contradiction...?

http://yorke.umd.edu/math410/HW4_Showing_the_rationals_are_dense_in_the_reals.htm

Forgive my stupidity, but I'm trying to reason something out.

Between any two rational numbers A and B, we have an uncountably infinite number of real numbers. Also, between any two real numbers C and D we have an uncountably infinite number of real numbers.

Now, between any two real numbers C and D, we have at least one rational number m0/n0 between C and D as demonstrated in flexmore's link above. For argument's sake, let's pick another real number E such that C < m0/n0 < E < D. Again, as per flexmore's link, there must exist a rational number m1/n1 between E and D. It seems we can repeat this process an uncountably infinite number of times, because there are an uncountably infinite number of "nested" real numbers for us to choose from, each with a corresponding rational number somewhere in the middle. This seems to imply that the number of rational numbers is the same as the number of nested real numbers, which is uncountably infinite.

I know that the rational numbers are countable, not uncountable, so something must be wrong here. Can anyone help me out?

Originally posted by PBE6
It seems we can repeat this process an uncountably infinite number of times

There's your problem. Each time you do this you are adding 1 more number. By definition you're going to end up with a countable number - because you're in effect counting them!

Originally posted by PBE6
Forgive my stupidity, but I'm trying to reason something out.

Between any two rational numbers A and B, we have an uncountably infinite number of real numbers. Also, between any two real numbers C and D we have an uncountably infinite number of real numbers.

Now, between any two real numbers C and D, we have at least one rational number m0/n0 between C a ...[text shortened]... are countable, not uncountable, so something must be wrong here. Can anyone help me out?

The set of sets of rational numbers does have an uncountable number of members (a simple proof with a mapping of all reals into that set: for each real number just take the set of all rationals smaller than that real number, each real will link directly to it's set).

on a different track:
When you propose repeating a process through an uncountable set , theory says you will not be able to step through it reaching all members ... you have not proposed a method of stepping through the process, so you are not forcing anything to be countable.